Dynamical study of phase fluctuations Dynamical study of phase fluctuations and their critical slowing down in and their critical slowing down in

amorphous superconducting filmsamorphous superconducting films

Wei LiuWei LiuThe Johns Hopkins UniversityThe Johns Hopkins University

Wei Liu, et al, Phys. Rev. B 84, 024511

(2011)

1

Acknowledgement Acknowledgement

N. Peter Armitage (JHU)

Rolando Valdes Aguilar (JHU)

Luke Bilbro(JHU)

Sambandamurthy Ganapathy(UB)

Minsoo Kim(UB)

22

OutlineOutline

Overview Broadband Corbino microwave

spectrometer InOx thin films Results and discussion Conclusion

33

OutlineOutline

Overview Broadband Corbino microwave

spectrometer InOx thin film Results and discussions Conclusion

44

Superconducting Superconducting fluctuationsfluctuations

Superconducting order Superconducting order parameter:parameter:

==eeii

• Amplitude Amplitude fluctuations: fluctuations: Ginzburg-Landau theoryGinzburg-Landau theory

• Phase fluctuations: Phase fluctuations: thermally generated free thermally generated free vorticesvortices

• Kosterlitz-Thouless-Kosterlitz-Thouless-Berezinskii phase Berezinskii phase transition: transverse transition: transverse phase fluctuations frozen phase fluctuations frozen out out

Ω/

Temperature (Kelvin)TKTB Tc0

Am

pli

tud

e F

luct

uat

ion

s

Ph

ase F

luctu

ati

on

s

Su

pe

rco

nd

uct

ivit

y

No

rmal

Sta

te

55

Kosterlitz-Thouless - Kosterlitz-Thouless - Berezinskii Berezinskii

Kosterlitz, Thouless: J. Phys. C: solid phys, Vol. 6 1973

Berezinskii, Sov. Phys. JETP 32 (1971) 493

From V. Vinokur

Ω/

Temperature (Kelvin)TKTB Tc0

Am

pli

tud

e F

luct

uat

ion

s

Ph

ase F

luctu

ati

on

s

Su

pe

rco

nd

uct

ivit

y

No

rmal

Sta

te

6

Universal resistance curveUniversal resistance curve

P. Minnhagen (1987)

77

Non linear I-V characteristicNon linear I-V characteristic

K. Epstein (1982)

88

Universal JumpUniversal Jump

McQueeny et al. (1984) He3-He4 mixtures of different proportions

Pproportional to superfluid density - Measured via torsion oscillator

99

Frequency Dependent Superfluid StiffnessFrequency Dependent Superfluid Stiffness

1010

ConclusionConclusion Unique system: continuous scan to measure Unique system: continuous scan to measure

complex conductivity down to 300 mK at complex conductivity down to 300 mK at microwave region; capable to perform finite microwave region; capable to perform finite frequency study on 2D quantum phase transition.frequency study on 2D quantum phase transition.

Superfluid stiffness acquires frequency Superfluid stiffness acquires frequency dependence at a transition temperature which is dependence at a transition temperature which is close to the universal jump value close to the universal jump value -consistent with Kosterlitz-Thouless-Berezinskii formalism.-consistent with Kosterlitz-Thouless-Berezinskii formalism.

Critical slowing down close to the phase transition Critical slowing down close to the phase transition and in general the applicability of a vortex and in general the applicability of a vortex plasma model above Tc. plasma model above Tc.

1111

OutlineOutline

Motivation Broadband Corbino microwave Broadband Corbino microwave

spectrometerspectrometer InOx thin film Results and discussions Conclusion

1212

Corbino Microwave SpectrometerCorbino Microwave Spectrometer

Broadband microwave spectroscopy has traditionally been difficult

Most measurements with microwave cavities, but they are limited to some particular frequencies

Our broadband microwave Corbino spectrometer can scan from 10MHz to 40GHz with 1Hz resolution down to 300mK

Measure both component of complex ‘optical’ response σ=σ1+iσ2 over a broad microwave frequency range

1313

Corbino SpectrometerCorbino Spectrometer

1414

OutlineOutline

Motivation Broadband Corbino microwave

spectrometer InOInOxx thin film thin film Results and discussion Conclusion

1515

Films prepared by e-gun evaporating high purity (99.999 %) In2O3 on clean 0.38mm thick 4.4mm*4.4mm Silicon substrate. High Tc at high resistance – 2.3K @ 7kW. Current films are 30nm thick morphologically homogeneous and amorphous.

Inherent disorder can be tuned by thermal annealing slightly above room temperature

InOInOxx film growth film growth

amorphous granular

(A) and (B) are AFM images of InOx samples grown at SUNY-Buffalo by varying growth conditions.

(C) Transmission electron diffraction image of an amorphous, homogeneous sample showing the non-crystalline nature of the film

1616

OutlineOutline

Motivation Broadband Corbino microwave

spectrometer InOx thin film Results and discussionResults and discussion Conclusion

1717

Tc0 is extracted using the Aslamazov-Larkin theory

for DC fluctuation superconductivity

(amplitude fluctuations).

The temperature scale at which Cooper pairs start

to form

Tc0 an energy scale in 2D, but not a phase transition

…

(x,t)ei(x,t)

Temperature (Kelvin)

Extracting TExtracting Tc0c0-The Cooper Paring -The Cooper Paring scalescale

1818

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0

Con

duct

ivity

806040200

Frequency

Real ConductivityImaginary Conductivity

Superconductor AC conductanceSuperconductor AC conductance

1919

AC Response of a AC Response of a SuperconductorSuperconductor

Canonical response of a superconductor at low T

Real and imaginary part of conductance plotted as a function of frequency for different temperatures

2020

Frequency Dependent Superfluid StiffnessFrequency Dependent Superfluid Stiffness

Superfluid density can be parameterized as a superfluid stiffness: Energy scale to twist superconducting phase e

1 23 4 5 6

Spin stiffness in discrete model.

2121

Kosterlitz-Thouless-Berezenskii Transition

In 2D static superfluid stiffness falls discontinuously to zero at temperature set

by superfluid stiffness itself. Thermal vortex/anti-vortex proliferation at TKTB.

Sup

erflu

id s

tiffn

ess

TKTB

Temperature

Universal jump in Superfluid (Phase) Universal jump in Superfluid (Phase) StiffnessStiffness

2222

Kosterlitz Thouless Berezenskii TransitionS

uper

fluid

stif

fnes

TKTB Tm

bare superfluid stiffness

=0

=inf

In 2D static superfluid stiffness survives at finite frequency (amplitude is still well defined). Finite frequency probes short length scale. If then system looks superconducting. Approaches ‘bare’ stiffness as gets big.

Temperature

increasing

Frequency Dependent Superfluid Stiffness Frequency Dependent Superfluid Stiffness ……

Probing length set by diffusion relation.

2323

Frequency Dependent Superfluid Stiffness …Frequency Dependent Superfluid Stiffness …

2424

Universal jump?

Tpredicted

Tcritical

Non-universaljump?

2525

Superconductor AC Superconductor AC ConductanceConductance

2626

Fisher-Widom Scaling Hypothesis

“Close to continuous transition, diverging length and time scales dominate response functions. All other lengths should be

compared to these”

Scaling AnalysisScaling Analysis

2727

Close to transition scaling forms are expected.

Data collapse with characteristic relaxation frequency (T) = 1/

Important! Since pre-factors are real, phase of S is also phase of !With = tan-1(2/1). should collapse with one parameter scaling.

Scaling in superconductorsScaling in superconductors

Functional form may look unusual, but it is

not. Drude model obeys this form.

All temperature dependencies enter through extracted and T from scaling

2828

Scaling in 2D superconductors: PhaseScaling in 2D superconductors: Phase

2929

Scaling in 2D superconductors: PhaseScaling in 2D superconductors: Phase

All temperature dependencies enter through extracted and T from scaling

3030

Scaling in 2D superconductors: Scaling in 2D superconductors: MagnitudeMagnitude

3131

t

Scaling in 2D superconductors: Scaling in 2D superconductors: MagnitudeMagnitude

3232

Characteristic fluctuation Characteristic fluctuation raterate

3333

Scaling in 2D superconductorsScaling in 2D superconductors

GHz and z = 1.58GHz and T’

3434

Vortex Activation?

our value of T’ is consistent with a reasonably small value of the vortex core energy

GHz and T’B. Halperin et al. J. Low Temp. Phys. 36, 599 (1979).

L. Benfatto et al. Phys. Rev. B 80,

3535

α is the ratio of is the votex core energy μ , to the votex core energy in the 2D XY model μXY

Vortex Activation?

We get 0.27K, which compares with estimate from T

0 approximately0.3 K

Within BCS one expects that: ~ T

0/8

3636

T0/8

ConclusionConclusion Unique system: continuous scan to measure Unique system: continuous scan to measure

complex conductivity down to 300 mK at complex conductivity down to 300 mK at microwave region; capable to perform finite microwave region; capable to perform finite frequency study on 2D quantum phase transition.frequency study on 2D quantum phase transition.

Superfluid stiffness acquires frequency Superfluid stiffness acquires frequency dependence at a transition temperature which is dependence at a transition temperature which is close to the universal jump value close to the universal jump value -consistent with Kosterlitz-Thouless-Berezinskii formalism.-consistent with Kosterlitz-Thouless-Berezinskii formalism.

Critical slowing down close to the phase transition Critical slowing down close to the phase transition and in general the applicability of a vortex and in general the applicability of a vortex plasma model above Tc. plasma model above Tc.

3737

383838

Scheme of sampleScheme of sample

Scheffler et al.

Superfluid (Phase) Stiffness …

Many of the different kinds of superconducting fluctuations can be viewed as disturbance in phase

field

Energy for deformation of any continuous elastic medium (spring, rubber, etc.) has a form that goes like square of generalized coordinate squared e.g. Hooke’s law

U = ½ kx2

3939

= sc phase q

Kosterlitz Thouless Berzenskii Kosterlitz Thouless Berzenskii TransitionTransition

Superfl

uid

sti

ffnes

TKTB Tm

bare superfluid density

w=0

w=inf

Temperature

increasing w

4040

Q: What about ‘normal’ electrons?

In principle there can be a contribution to 2 from thermally excited electrons

and above gap excitations.

Rough estimate, using Drude relations and approximate numbers …

A: Due to strong scattering ‘normal’ electrons give completely insignificant contribution @ our frequencies

0.001

0.01

0.1

1

Imag

inar

y C

ondu

ctiv

ity

12 3 4 5 6 7 8 9

102 3 4 5 6 7 8 9

100

Frequency

=32

=16

=8

=5

=3

=inf

0.001

0.01

0.1

1

Imag

inar

y C

ondu

ctiv

ity

12 3 4 5 6 7 8 9

102 3 4 5 6 7 8 9

100

Frequency

=32

=16

=8

=5

=3

=inf

4141

Superconductor AC Superconductor AC ConductanceConductance

Close to transition scaling forms for the conductivity are

expected *.

Data collapse in terms of a characteristic relaxation

frequency (T) =

* Fisher, Fisher, Huse PRB, 1991

4242

434343

Sigma2Sigma2

Superconductor AC Superconductor AC ConductanceConductance

4444

454545

References:References:

1.1. Marc Scheffler, Broadband Microwave Spectroscopy on Correlated Electrons, Dissertation, Marc Scheffler, Broadband Microwave Spectroscopy on Correlated Electrons, Dissertation, Universität Stuttgart, Stuttgart,2004Universität Stuttgart, Stuttgart,2004

2.2. Riley Crane, Probing the Bose Solid: A finite frequency study of the magnetic field-tuned Riley Crane, Probing the Bose Solid: A finite frequency study of the magnetic field-tuned superconductor-insulator transition in two-dimensions, Dissertation, UCLA, CA, 2006superconductor-insulator transition in two-dimensions, Dissertation, UCLA, CA, 2006

3.3. James Clay Booth, Novel Measurements of the Frequency Dependent Microwave Surface James Clay Booth, Novel Measurements of the Frequency Dependent Microwave Surface Impedance of Cuprate Thin Film Superconductors, Dissertation, university of Maryland, 1996Impedance of Cuprate Thin Film Superconductors, Dissertation, university of Maryland, 1996

4.4. R. W. Crane, N. P. Armitage, A. Johansson, G. Sambandamurthy, D. Shahar, and G. Gruner, R. W. Crane, N. P. Armitage, A. Johansson, G. Sambandamurthy, D. Shahar, and G. Gruner, SSurvival of superconducting correlations across the two-dimensional superconductor-insulator urvival of superconducting correlations across the two-dimensional superconductor-insulator transition: A finite-frequency study transition: A finite-frequency study ,, Phys. Rev. B 75, 184530 (2007) Phys. Rev. B 75, 184530 (2007)

5.5. R. W. Crane, N. P. Armitage, A. Johansson, G. Sambandamurthy, D. Shahar, and G. Gruner, R. W. Crane, N. P. Armitage, A. Johansson, G. Sambandamurthy, D. Shahar, and G. Gruner, FFluctuations, dissipation, and nonuniversal superfluid jumps in two-dimensional luctuations, dissipation, and nonuniversal superfluid jumps in two-dimensional superconductorssuperconductors,, Phys. Rev. B 75, 094506 (2007) Phys. Rev. B 75, 094506 (2007)

6.6. Martin Dressel and George Gruner, Electrodynamics of Solids: Optical Properties of Electrons in Martin Dressel and George Gruner, Electrodynamics of Solids: Optical Properties of Electrons in Matter (Cambridge University Press, Cambridge, 2002).Matter (Cambridge University Press, Cambridge, 2002).

7.7. Marc Scheffler and Martin Dressel, Marc Scheffler and Martin Dressel, BBroadband microwave spectroscopy in Corbino geometry for roadband microwave spectroscopy in Corbino geometry for temperatures down to 1.7 Ktemperatures down to 1.7 K,, Rev. Sci. Instrum. 76, 074702 (2005) Rev. Sci. Instrum. 76, 074702 (2005)

8.8. S. M. Girvin, Duality in Perspective, Science 25, Vol. 274. no. 5287, pp. 524 - 525 (1996)S. M. Girvin, Duality in Perspective, Science 25, Vol. 274. no. 5287, pp. 524 - 525 (1996)9.9. J. C. Booth, Dong Ho Wu, and Steven M. Anlage, J. C. Booth, Dong Ho Wu, and Steven M. Anlage, AA broadband method for the measurement of broadband method for the measurement of

the surface impedance of thin films at microwave frequenciesthe surface impedance of thin films at microwave frequencies,, Rev. Sci. Instrum. 65, 2082 Rev. Sci. Instrum. 65, 2082 (1994)(1994)

10.10. Marc Scheffler, Serife Kilic, and Martin Dressel, Strip-shaped samples in a microwave Corbino Marc Scheffler, Serife Kilic, and Martin Dressel, Strip-shaped samples in a microwave Corbino spectrometer, Rev. Sci. Instrum 78, 086106 (2007) spectrometer, Rev. Sci. Instrum 78, 086106 (2007)

11.11. James C. Booth, Dong-Ho Wu, and Steven M. Anlage, Measurements of the Frequency James C. Booth, Dong-Ho Wu, and Steven M. Anlage, Measurements of the Frequency Dependent Microwave Fluctuation Conductivity of Cuprate Thin Film Superconductors, Dependent Microwave Fluctuation Conductivity of Cuprate Thin Film Superconductors, Fluctuation Phenomena in High Temperature Superconductors, (Kluwer, Dordrecht, 1997), edited Fluctuation Phenomena in High Temperature Superconductors, (Kluwer, Dordrecht, 1997), edited by Marcel Ausloos and Andrei A. Varlamov, pp.151 - 178.by Marcel Ausloos and Andrei A. Varlamov, pp.151 - 178.

12.12. Haruhisa Kitano, Takeyoshi Ohashi and Atsutaka Maeda, Broadband method for precise Haruhisa Kitano, Takeyoshi Ohashi and Atsutaka Maeda, Broadband method for precise microwave spectroscopy of superconducting thin films near critical temperature, microwave spectroscopy of superconducting thin films near critical temperature, arxiv:0806.1421v1 (2008)arxiv:0806.1421v1 (2008)

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